\newproblem{lay:1_4_35}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.4.35}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $A$ be a $5\times 3$ matrix, let $\mathbf{y}$ be a vector in $\mathbb{R}^3$, and let $\mathbf{z}$ be a vector in $\mathbb{R}^5$.
	Suppose $A\mathbf{y}=\mathbf{z}$. What fact allows you to conclude that the system $A\mathbf{x}=5\mathbf{z}$ is consistent?
}{
  % Solution
	If $A\mathbf{y}=\mathbf{z}$, then multiplying the equation by 5 we get
	\begin{center}
		$5(A\mathbf{y})=5\mathbf{z}$
	\end{center}
	Using the properties of scalar, matrix and vector multiplications, we may rearrange the equation as
	\begin{center}
		$A(5\mathbf{y})=5\mathbf{z}$
	\end{center}
	Now, simply calling $\mathbf{x}=5\mathbf{y}$ we get the equation proposed in the problem:
	\begin{center}
		$A\mathbf{x}=5\mathbf{z}$
	\end{center}
	whose solution is obviously $\mathbf{x}=5\mathbf{y}$.
}
\useproblem{lay:1_4_35}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
